Question

Reflect the graph of $y=\coth \left(x\right)$ in the line y= x to obtain the graph of $y$= ${\coth }^{-1}\left(x\right)$

Answer 1

$y=\coth x$

$=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$

$\because e^x-e^{-x}\ne 0e^{2x}\ne 1$

$\Rightarrow x\ne 0$

Hence $x\in R-\left\{0\right\}$

$y={\coth }^{-1}x$

$=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right)$

Here $x\ne 1$

also, $\frac{x+1}{x-1}>0$

$x>-1$ also, $x\ne 0$

Hence graph will be as shown above.